Determine how many solutions exist for the system of equations. ${8x+2y = 6}$ ${y = -5+3x}$
Solution: Convert both equations to slope-intercept form: ${8x+2y = 6}$ $8x{-8x} + 2y = 6{-8x}$ $2y = 6-8x$ $y = 3-4x$ ${y = -4x+3}$ ${y = -5+3x}$ ${y = 3x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+3}$ ${y = 3x-5}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.